20140602-UPMC.tex

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\title{Next-generation multigridding: adaptivity and communication avoidance}
\author{{\bf Jed Brown} \texttt{jedbrown@mcs.anl.gov} (ANL and CU Boulder) \\
  \quad Mark Adams (LBL), Matt Knepley (UChicago)
}

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% {
%   Mathematics and Computer Science Division \\ Argonne National Laboratory
% }

\date{UPMC, 2014-06-02}

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\subject{Talks}


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\lstset{language=C}
\normalem

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Plan: ruthlessly eliminate communication}
  \begin{block}{Why?}
    \begin{itemize}
    \item Local recovery despite global coupling
    \item Tolerance for high-frequency load imbalance
      \begin{itemize}
      \item From irregular computation or hardware error correction
      \end{itemize}
    \item More scope for dynamic load balance
    \end{itemize}
  \end{block}
  \begin{block}{Requirements}
    \begin{itemize}
    \item Must retain optimal convergence with good constants
    \item Flexible, robust, and debuggable
    \end{itemize}
  \end{block}
\end{frame}

\section{$\tau$-adaptivity and multigrid compression}
\begin{frame}[fragile]{Multigrid Preliminaries}
  \begin{figure}
    \centering
    \begin{tikzpicture}
      [>=stealth,
      every node/.style={inner sep=2pt},
      restrict/.style={thick},
      prolong/.style={thick},
      mglevel/.style={rounded rectangle,draw=blue!50!black,fill=blue!20,thick,minimum size=4mm},
      ]
      \begin{scope}\scriptsize
        \newcommand\mgdx{4.0em}
        \newcommand\mgdy{4.0em}
        \newcommand\mgl[1]{(pow(2,#1+1))}
        \newcommand\mgloc[4]{(#1 + #4*\mgdx*#3,#2 + \mgdy*#3)}

        \newcommand\mghx{0.9*\mgdx}
        \newcommand\mghy{0.9*\mgdy}

        \draw[shift=\mgloc{0*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{3},
        ystep=\mghy/\mgl{3}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);

        \draw[shift=\mgloc{1*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{2},
        ystep=\mghy/\mgl{2}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);

        \draw[shift=\mgloc{2*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{1},
        ystep=\mghy/\mgl{1}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);


        \draw[shift=\mgloc{3*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{0},
        ystep=\mghy/\mgl{0}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);
      \end{scope}
    \end{tikzpicture}
    \label{fig:levels}
  \end{figure}
  \textbf{Multigrid} is an $O(n)$ method for solving algebraic problems by defining a hierarchy of scale.
  A multigrid method is constructed from:
  \begin{enumerate}
  \item a series of discretizations
    \begin{itemize}
    \item coarser approximations of the original problem
    \item constructed algebraically or geometrically
    \end{itemize}
  \item intergrid transfer operators
    \begin{itemize}
    \item residual restriction $I_h^H$ (fine to coarse)
    \item state restriction $\hat I_h^H$ (fine to coarse)
    \item partial state interpolation $I_H^h$ (coarse to fine, `prolongation')
    \item state reconstruction $\mathbb{I}_H^h$ (coarse to fine)
    \end{itemize}
  \item Smoothers ($S$)
    \begin{itemize}
    \item correct the high frequency error components
    \item Richardson, Jacobi, Gauss-Seidel, etc.
    \item Gauss-Seidel-Newton or optimization methods
    \end{itemize}
  \end{enumerate}
\end{frame}
\input{slides/MG/TauFAS.tex}

\begin{frame}{$\tau$ corrections}
  \begin{figure}
  \centering
  \begin{subfigure}[b]{0.18\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressTrim}
    %\caption{Initial solution.}\label{fig:elast-initial}
  \end{subfigure} ~
  \begin{subfigure}[b]{0.18\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressShearTrim}
    %\caption{Increment.}\label{fig:elast-increment}
  \end{subfigure} ~
  \begin{subfigure}[b]{0.28\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressErrorNoTauTrim}
    %\caption{Smoothed error without $\tau$.}\label{fig:elast-error-notau}
  \end{subfigure} ~
  \begin{subfigure}[b]{0.28\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressErrorTauTrim}
    %\caption{Smoothed error with $\tau$.}\label{fig:elast-error-tau}
  \end{subfigure}
  \begin{itemize}
  \item Plane strain elasticity, $E=1000,\nu=0.4$ inclusions in $E=1,\nu=0.2$ material, coarsen by $3^2$.
  \item Solve initial problem everywhere and compute $\tau_h^H = A^H \hat I_h^H u^h - I_h^H A^h u^h$
  \item Change boundary conditions and solve FAS coarse problem
    \begin{equation*}
      N^H \acute u^H = \underbrace{I_h^H \acute f^h}_{\acute f^H} + \underbrace{N^H \hat I_h^H \tilde u^h - I_h^H N^h \tilde u^h}_{\tau_h^H}
    \end{equation*}
  \item Prolong, post-smooth, compute error $e^h = \acute u^h - (N^h)^{-1} \acute f^h$
  \item<2> \alert{Coarse grid \emph{with $\tau$} is nearly $10\times$ better accuracy}
  \end{itemize}
  % \caption{Plane strain elasticity, $E=1000,\nu=0.4$ inclusions in $E=1,\nu=0.2$ material.  2-level multigrid with coarsening factor of $3^2$.
  %   Panes (a) and (b) show the deformed body colored by strain.
  %   The initial problem of compression by 0.2 from the right is solved (a) and $\tau = A^H \hat I_h^H u^h - I_h^H A^h u^h$ is computed.
  %   Then a shear increment of 0.1 in the $y$ direction is added to the boundary condition, and the coarse-level problem is resolved, interpolated to the fine-grid, and a post-smoother is applied.
  %   When the coarse problem is solved without a $\tau$ correction (c), the displacement error is nearly $10\times$ larger than when $\tau$ is included in the right hand side of the coarse problem (d).
  % }\label{fig:tau-valid}
  % ./ex49 -mx 90 -my 90 -da_refine_x 3 -da_refine_y 3 -elas_ksp_converged_reason -elas_ksp_rtol 1e-8 -no_view -c_str 3 -sponge_E0 1 -sponge_E1 1e3 -sponge_nu0 0.4 -sponge_nu1 0.2 -sponge_t 3 -sponge_w 9 -u_o vtk:ex49_sol.vts -use_nonsymbc -elas_pc_type mg -elas_pc_mg_levels 2 -elas_pc_mg_galerkin -tau1_o vtk:ex49_tau1.vts -tau2_o vtk:ex49_tau2.vts -taudiff_o vtk:ex49_taudiff.vts -u2_o vtk:ex49_sol2.vts -u2c_o vtk:ex49_sol2c.vts -u3_o vtk:ex49_sol3.vts -u4_o vtk:ex49_sol4.vts -u2err_o vtk:ex49_sol2err.vts -u3err_o vtk:ex49_sol3err.vts -u3c_o vtk:ex49_sol3c.vts -tau3_o vtk:ex49_tau3.vts
\end{figure}
\end{frame}

\begin{frame}{$\tau$ adaptivity for heterogeneous media}
  \begin{itemize}
  \item Applications
    \begin{itemize}
    \item Geo: reservoir engineering, lithosphere dynamics (subduction, rupture/fault dynamics)
    \item carbon fiber, biological tissues, fracture
    \item Conventional adaptivity fails
    \end{itemize}
  \item Traditional adaptive methods fail
    \begin{itemize}
    \item Solutions are not ``smooth''
    \item Cannot build accurate coarse space without scale separation
    \end{itemize}
  \item $\tau$ adaptivity
    \begin{itemize}
    \item Fine-grid work needed everywhere at first
    \item Then $\tau$ becomes accurate in nearly-linear regions
    \item Only visit fine grids in ``interesting'' places: active nonlinearity, drastic change of solution
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Comparison to nonlinear domain decomposition}
  \begin{itemize}
  \item ASPIN (Additive Schwarz preconditioned inexact Newton) \\
    \begin{itemize}
    \item Cai and Keyes (2003)
    \item More local iterations in strongly nonlinear regions
    \item Each nonlinear iteration only propagates information locally
    \item Many real nonlinearities are activated by long-range forces
      \begin{itemize}
      \item locking in granular media (gravel, granola)
      \item binding in steel fittings, crack propagation
      \end{itemize}
    \item Two-stage algorithm has different load balancing
      \begin{itemize}
      \item Nonlinear subdomain solves
      \item Global linear solve
      \end{itemize}
    \end{itemize}
  \item $\tau$ adaptivity
    \begin{itemize}
    \item Minimum effort to communicate long-range information
    \item Nonlinearity sees effects as accurate as with global fine-grid feedback
    \item Fine-grid work always proportional to ``interesting'' changes
    \end{itemize}
  \end{itemize}
\end{frame}

\subsection{Reducing communication and memory bandwidth}
\input{slides/MG/LowComm.tex}
\begin{frame}{Segmental refinement: no horizontal communication}
  \begin{itemize}
  \item 27-point second-order stencil, manufactured analytic solution
  \item 5 SR levels: $16^3$ cells/process local coarse grid
  \item $\text{Overlap} = \text{Base} + (L-\ell) \text{Increment}$
    \begin{itemize}
    \item Implementation requires even number of cells---round down.
    \end{itemize}
  \item FMG with $V(2,2)$ cycles
  \end{itemize}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \begin{table}\small
        \centering\caption{$\norm{e_{SR}}_\infty / \norm{e_{FMG}}_\infty$}\label{tab:sr-error}
        \begin{tabular}{l rrr}
          \toprule
          & \multicolumn{3}{c}{Base} \\
          Increment & 1 & 2 & 3 \\
          \midrule
          1 & {\color{red} 1.59} & {\color{red} 2.34} & 1.00 \\
          2 & 1.00 & 1.00 & 1.00 \\
          3 & 1.00 & 1.00 & 1.00 \\
          \bottomrule
        \end{tabular}
      \end{table}
    \end{column}
    \begin{column}{0.6\textwidth}
      \includegraphics[width=\textwidth]{figures/MG/weak_scaling_edison-eps-converted-to.pdf}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}{Reducing memory bandwidth}
  \includegraphics[width=\textwidth]{figures/MG/SRMGWindow}
  \begin{itemize}
  \item Sweep through ``coarse'' grid with moving window
  \item Zoom in on new slab, construct fine grid ``window'' in-cache
  \item Interpolate to new fine grid, apply pipelined smoother ($s$-step)
  \item Compute residual, accumulate restriction of state and residual into coarse grid, expire slab from window
  \end{itemize}
\end{frame}

\begin{frame}{Arithmetic intensity of sweeping visit}
  \begin{itemize}
  \item Assume 3D cell-centered, 7-point stencil
  \item 14 flops/cell for second order interpolation
  \item $\ge 15$ flops/cell for fine-grid residual or point smoother
  \item 2 flops/cell to enforce coarse-grid compatibility
  \item 2 flops/cell for plane restriction
  \item assume coarse grid points are reused in cache
  \item Fused visit reads $u^H$ and writes $\hat I_h^H u^h$ and $I_h^H r^h$
  \item Arithmetic Intensity
    \begin{equation}
      \frac{{\overbrace{15}^{\text{interp}}} + {\overbrace{2\cdot (15+2)}^{\text{compatible relaxation}}} + \overbrace{2\cdot 15}^{\text{smooth}} + \overbrace{15}^{\text{residual}} + \overbrace{2}^{\text{restrict}}}{3 \cdot \texttt{sizeof(scalar)} / \underbrace{2^3}_{\text{coarsening}}} \gtrsim 30
    \end{equation}
  \item Still $\gtrsim 10$ with non-compressible fine-grid forcing
  \end{itemize}
\end{frame}

\begin{frame}{Regularity}
  Accuracy of recovery depends on operator regularity
  \begin{itemize}
  \item Even with regularity, we can only converge up to discretization error, unless we add a \emph{consistent} fine-grid residual evaluation
  \item Visit fine grid with some overlap, but patches do not agree exactly in overlap
  \item Need decay length for high-frequency error components (those that restrict to zero) that is bounded with respect to grid size
  \item Required overlap $J$ is proportional to the number of cells to cover decay length
  \item Can enrich coarse space along boundary, but causes loss of coarse-grid sparsity
  \item Brandt and Diskin (1994) has two-grid LFA showing $J \lesssim 2$ is sufficient for Laplacian
  \item With $L$ levels, overlap $J(k)$ on level $k$,
    \begin{equation*}
      2J(k) \ge s (L-k+1)
    \end{equation*}
    where $s$ is the smoothness order of the solution or the discretization order (whichever is smaller)
  \end{itemize}
\end{frame}

\subsection{Local recovery and postprocessing}
\begin{frame}{Basic resilience strategy}
  \begin{tikzpicture}
    [scale=0.8,every node/.style={scale=0.8},
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    control/.style={rectangle,rounded corners,draw=blue!50!black,fill=blue!20,thick,minimum width=5em},
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    rightbrace/.style={decorate,decoration={brace,amplitude=1ex,raise=4pt}},
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    \scriptsize
    \node[control,minimum width=8em] (progcontrol) {control};
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    \node[above=0pt of progbox.north,anchor=south] {\textbf{program $n=0$}};

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    \node[essential,below=2pt of storecontrol.south] (storeessential) {essential};
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    \node[control,right=7em of storecontrol] (reccontrol) {control};
    \node[essential,below=2pt of reccontrol.south] (recessential) {essential};
    \node[statebox,fit=(reccontrol)(recessential)] (recbox) {};
    \node[above=0pt of recbox.north,anchor=south] {\textbf{restored $n=0$}};

    \node[control,right=6em of reccontrol] (donecontrol) {control};
    \node[essential,below=2pt of donecontrol.south] (doneessential) {essential};
    \node[ephemeral,below=2pt of doneessential.south] (doneephemeral) {ephemeral};
    \node[statebox,fit=(donecontrol)(doneephemeral)] (donebox) {};
    \node[above=0pt of donebox.north,anchor=south] {\textbf{recovered $n=N$}};

    \draw[decorate,decoration={brace,amplitude=1ex,raise=4pt}] ($(progcontrol.north east) + (3pt,0)$) -- ($(progephemeral.north east) + (3pt,0)$) node[midway,xshift=1ex] (progbrace) {};
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    \draw[rightbrace] ($(storecontrol.north east) + (4pt,0)$) -- ($(storeessential.south east) + (4pt,0)$) node[midway,xshift=1ex] (storerbrace) {};
    \draw[->,shorten >=4pt,shorten <=4pt] (progbrace) -- (storebrace) node[midway,above] (midarrow) {MPI/BLCR};

    \node[below=1.4em of midarrow,essential,draw=red!50!gray!70,fill=red!10] (coarserun) {};
    \draw[->,dashed,shorten >=14pt,shorten <=4pt] (coarserun) |- (storecoarse) node [near start,below,yshift=-3pt] {\scriptsize $n=1,2,\dotsc,N$};
    \draw[->,shorten >=4pt,shorten <=4pt] (storerbrace) -- (recbox.west) node[midway,above,text width=5em,align=center] (midarrow) {restart failed ranks};
    \draw[->,shorten >=5pt,shorten <=4pt] (recessential.east) -- (doneessential) node[midway,above,text width=5em,align=center] (fmgrecover) {FMG recovery};
    \draw[->,dashed,shorten >=1pt,shorten <=3pt] ($(storecoarse.east) + (1em,0)$) -| (fmgrecover) node[midway,below,xshift=-1em] {\scriptsize $n=1,2,\dotsc,N$};
    \draw[->,dashed,shorten >=3pt,shorten <=3pt] (donecontrol.east) -| ($(donecontrol.east) + (3ex,0)$) |- (doneephemeral.east) node[midway,right,text width=4em] {\cverb|malloc| at $n=0$};
  \end{tikzpicture}
\begin{description}
\item[control] contains program stack, solver configuration, etc.
\item[essential] program state that cannot be easily reconstructed: time-dependent solution, current optimization/bifurcation iterate
\item[ephemeral] easily recovered structures: assembled matrices, preconditioners, residuals, Runge-Kutta stage solutions
\end{description}
\begin{itemize}
\item Essential state at time/optimization step $n$ is \alert{inherently globally coupled} to step $n-1$ (otherwise we could use an explicit method)
\item \emph{Coarse} level checkpoints are orders of magnitude smaller, but allow rapid recovery of essential state
\item FMG recovery needs only \alert{nearest neighbors}
\end{itemize}
\end{frame}

\begin{frame}[fragile]{Multiscale compression and recovery using $\tau$ form}
   \begin{tikzpicture}
    [scale=0.7,every node/.style={scale=0.7},
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    restrict/.style={thick,double},
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      \node[mglevel] (cpdown0) at \mgloc{0}{0}{1}{-1} {\mglevelcp};
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      \node[mglevel] (cp1up1) at \mgloc{4em}{0}{2}{1} {$\mglevelcp+1$};
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      \draw[->,restrict] (cp1down0) -- (cpdown0);
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      \draw[->,prolong,dashed] (finem1up1) -- (fine1);

      \node[checkpoint] at (4em + \mgdx*4,\mgdy) (cp) {CP};
      \draw[>->,cprestrict] (fine1) -- node[below,sloped] {Restrict} (cp);

      \node[left=\mgdx of fine0] (bnanchor) {};
      \node[control,fill=red!20] at (1.1*\mgdx,3*\mgdy) {Solve $F(u^n;b^n) = 0$};
      \node[mglevel,right=of fine1] (finedt) {next solve};
      \draw[->, >=stealth, control] (fine1) to[out=20,in=170] node[above] {$b^{n+1}(u^n,b^n)$} (finedt);
      \draw[->, >=stealth, control] (bnanchor) to[out=45,in=155] node[above] {$b^n$} (fine0);

      % Recovery process
      \begin{scope}[xshift=8*\mgdx]
        \node[checkpoint] (rcp) at \mgloc{0}{0}{0}{0} {CP};
        \node[mglevel] (r0a) at \mgloc{0}{\mgdy}{0}{0} {CR};
        \node[mglevel] (r1a) at \mgloc{0}{\mgdy}{1}{1} {};
        \node[mglevel] (r0b) at \mgloc{2*\mgdx}{\mgdy}{0}{0} {CR};
        \node[mglevel] (r1b) at \mgloc{2*\mgdx}{\mgdy}{1}{1} {};
        \node[mglevel] (r2b) at \mgloc{2*\mgdx}{\mgdy}{2}{1} {\mglevelfine};
        \node[mglevel] (r1c) at \mgloc{6*\mgdx}{\mgdy}{1}{-1} {};
        \node[mglevel] (r0d) at \mgloc{6*\mgdx}{\mgdy}{0}{0} {CR};
        \node[mglevel] (r1d) at \mgloc{6*\mgdx}{\mgdy}{1}{1} {};
        \node[mglevel] (r2d) at \mgloc{6*\mgdx}{\mgdy}{2}{1} {\mglevelfine};

        \draw[-,prolong,green!50!black] (rcp) -- (r0a);
        \draw[->,prolong] (r0a) -- (r1a);
        \draw[->,restrict] (r1a) -- (r0b);
        \draw[->,restrict] (r0b) -- (r1b);
        \draw[->,restrict,dashed] (r1b) -- (r2b);
        \draw[->,restrict,dashed] (r2b) -- (r1c);
        \draw[->,restrict] (r1c) -- (r0d);
        \draw[->,restrict] (r0d) -- (r1d);
        \draw[->,restrict,dashed] (r1d) -- (r2d);

        \foreach \smooth in {finem1down0, cp1down0, cpdown0, coarser0,
          cpup1, cp1up1, finem1up1,
          r0b,r1c,r0d,r1d} {
          \node[above left=-5pt of \smooth.west,tau] {$\tau$};
        }
        \node[rectangle,fill=none,draw=green!50!black,thick,fit=(rcp)(r2d)] (recoverbox) {};
        \node[rectangle,draw=green!50!black,fill=green!20,thick,minimum size=6mm,above={0cm of recoverbox.south east},anchor=south east] (recover) {FMG Recovery};
      \end{scope}
      \node (notation) at (\mgdx,5*\mgdy) {
        \begin{minipage}{18em}\small\sf
          \begin{itemize}\addtolength{\itemsep}{-5pt}
          \item checkpoint converged coarse state
          \item recover using FMG anchored at $\mglevelcp+1$
          \item needs only $\mglevelcp$ neighbor points
          \item $\tau$ correction is local
          \end{itemize}
        \end{minipage}
      };
    \end{scope}
  \end{tikzpicture}
  \begin{itemize}
  \item Normal multigrid cycles visit all levels moving from $n \to n+1$
  \item FMG recovery only accesses levels finer than $\ell_{CP}$
  \item Only failed processes and neighbors participate in recovery
  \item Lightweight checkpointing for transient adjoint computation
  \item Postprocessing applications, e.g., in-situ visualization at high temporal resolution in part of the domain
  \end{itemize}
\end{frame}

\begin{frame}{First-order cost model for FAS resilience}
  Extend first-order locality-unaware model of Young (1974):
  \begin{description}
  \item[$\timeW$] time to write a heavy fine-grid checkpointed state
  \item[$\timeR$] time to read back lost state
  \item[$R$] fraction of forward simulation needed for recomputation from a saved state
  \item[$P$] the heavy checkpoint interval
  \item[$M$] mean time to failure
  \end{description}
  Neglect cost of I/O for lightweight coarse-grid checkpoints
  \begin{equation*}\label{eq:overhead}
    \text{Overhead} = 1 - \text{AppUtilization} = \underbrace{\frac{\timeW}{P}}_{\text{writing}}
    + \underbrace{\frac{\timeR}{M}}_{\text{reading after failure}}
    + \underbrace{\frac{R P}{2M}}_{\text{recomputation}}
  \end{equation*}
  Minimized for a heavy checkpointing interval $P = \sqrt{2 M \timeW / R}$
  \begin{equation*}\label{eq:minoverhead}
    \text{Overhead}^* = \sqrt{2 \timeW R / M} + \timeR / M
    % $ \text{Overhead}^* = \sqrt{\frac{2 \timeW R}{M}} + \frac{\timeR}{M} $,
  \end{equation*}
  where the first term is always larger than the second.
  Conventional checkpointing schemes store only fine-grid state, thus $R=1$ (recovery costs the same as initial computation).
\end{frame}

\begin{frame}{Other uses}
  \begin{itemize}
  \item Transient adjoints
    \begin{itemize}
    \item Adjoint model runs backward-in-time, needs state from solution of forward model
    \item Status quo: hierarchical checkpointing
    \item Memory-constrained and requires computing forward model multiple times
    \item If forward model is stiff, each step has global dependence
    \item Compression via $\tau$-FAS accelerates recomputation, can be local
    \end{itemize}
  \item Visualization and analysis
    \begin{itemize}
    \item Targeted visualization in small part of domain
    \item Interesting features emergent so can't predict where to look
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Outlook}
  \begin{itemize}
  \item System software challenge: Partial restart interfaces
    \begin{itemize}
    \item Low-overhead recruitment protocol via \texttt{MPI\_Ibarrier}
    \item Can we ``reconnect'' MPI communicators?
    \item Can a library return a solution on a different communicator?
    \end{itemize}
  \item $\tau$-FAS adaptivity and compression
    \begin{itemize}
    \item Coarse-centric restructuring is a major interface change
    \item Worthwhile for resilience and to better use hardware
    \item Nonlinear smoothers (and discretizations)
    \item Dynamic load balancing
    \item Reliability of error estimates for refreshing $\tau$
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{HPGMG-FE: a finite-element FAS benchmark}
  \begin{itemize}
  \item \url{https://hpgmg.org}
  \item Unassembled $Q_2$ elements, FMG solve
  \item Scales at 23\% of peak on Edison (Cray XC-30)
  \item 10x faster than assembled elements, less memory used
  \end{itemize}
  \centering
  \includegraphics[width=0.75\textwidth]{figures/MG/hpgmgfe-edison-vesta.png}
  %\includegraphics[width=0.75\textwidth]{figures/MG/fefas.png}
\end{frame}

\begin{frame}
  \centering
  \includegraphics[width=0.75\textwidth]{figures/hpgmg-fv-20140515-dof.png}
\end{frame}

\end{document}